## Tracking rotational diffusion of colloidal clusters |

Optics Express, Vol. 19, Issue 18, pp. 17189-17202 (2011)

http://dx.doi.org/10.1364/OE.19.017189

Acrobat PDF (1207 KB)

### Abstract

We describe a novel method of tracking the rotational motion of clusters of colloidal particles. Our method utilizes rigid body transformations to determine the rotations of a cluster and extends conventional proven particle tracking techniques in a simple way, thus facilitating the study of rotational dynamics in systems containing or composed of colloidal clusters. We test our method by measuring dynamical properties of simulated Brownian clusters under conditions relevant to microscopy experiments. We then use the technique to track and describe the motions of a real colloidal cluster imaged with confocal microscopy.

© 2011 OSA

## 1. Introduction

1. A. Kose, M. Ozaki, K. Takano, Y. Kobayashi, and S. Hachisu, “Direct observation of ordered latex suspension by metallurgical microscope,” J. Colloid Interface Sci. **44**, 330–338 (1973). [CrossRef]

8. E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, “Three-dimensional direct imaging of structural relaxation near the colloidal glass transition,” Science **287**, 627–631 (2000). [CrossRef] [PubMed]

9. J. R. Savage, D. W. Blair, A. J. Levine, R. A. Guyer, and A. D. Dinsmore, “Imaging the sublimation dynamics of colloidal crystallites,” Science **314**, 795–798 (2006). [CrossRef] [PubMed]

14. J. Hernández-Guzmán and E. R. Weeks, “The equilibrium intrinsic crystal-liquid interface of colloids,” Proc. Natl. Acad. Sci. U.S.A. **106**, 15198–15202 (2009). [CrossRef] [PubMed]

15. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

16. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. **40**, 4152–4159 (2001). [CrossRef]

17. E. R. Dufresne, T. M. Squires, M. P. Brenner, and D. G. Grier, “Hydrodynamic coupling of two Brownian spheres to a planar surface,” Phys. Rev. Lett. **85**, 3317–3320 (2000). [CrossRef] [PubMed]

21. D. Chen, D. Semwogerere, J. Sato, V. Breedveld, and E. R. Weeks, “Microscopic structural relaxation in a sheared supercooled colloidal liquid,” Phys. Rev. E **81**, 011403 (2010). [CrossRef]

22. S. Martin, M. Reichert, H. Stark, and T. Gisler, “Direct observation of hydrodynamic rotation-translation coupling between two colloidal spheres,” Phys. Rev. Lett. **97**, 248301 (2006). [CrossRef]

23. S. M. Anthony, L. Hong, M. Kim, and S. Granick, “Single-particle colloid tracking in four dimensions,” Langmuir **22**, 9812–9815 (2006). [CrossRef] [PubMed]

24. P. J. Yunker, K. Chen, Z. Zhang, W. G. Ellenbroek, A. J. Liu, and A. G. Yodh, “Rotational and translational phonon modes in glasses composed of ellipsoidal particles,” Phys. Rev. E **83**, 011403 (2011). [CrossRef]

27. D. Mukhija and M. J. Solomon, “Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy,” J. Colloid Interface Sci. **314**, 98–106 (2007). [CrossRef] [PubMed]

28. V. N. Manoharan, M. T. Elsesser, and D. J. Pine, “Dense packing and symmetry in small clusters of microspheres,” Science **301**, 483–487 (2003). [CrossRef] [PubMed]

29. M. T. Elsesser, A. D. Hollingsworth, K. V. Edmond, and D. J. Pine, “Large core-shell poly(methyl methacrylate) colloidal clusters: synthesis, characterization, and tracking,” Langmuir **27**, 917–927 (2011). [CrossRef]

30. A. van Blaaderen, “Chemistry: colloidal molecules and beyond,” Science **301**, 470–471 (2003). [CrossRef]

32. S. C. Glotzer and M. J. Solomon, “Anisotropy of building blocks and their assembly into complex structures,” Nature Mater. **6**, 557–562 (2007). [CrossRef]

15. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

16. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. **40**, 4152–4159 (2001). [CrossRef]

27. D. Mukhija and M. J. Solomon, “Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy,” J. Colloid Interface Sci. **314**, 98–106 (2007). [CrossRef] [PubMed]

33. S. M. Anthony, M. Kim, and S. Granick, “Translation-rotation decoupling of colloidal clusters of various symmetries,” J. Chem. Phys. **129**, 244701 (2008). [CrossRef]

## 2. Calculating rotations

34. J. Challis, “A procedure for determining rigid body transformation parameters,” J. Biomech. **28**, 733–737 (1995). [CrossRef] [PubMed]

### 2.1. Challis’ procedure for coordinate transformations

**E**is the identity matrix and det () denotes the determinant.

*n*points,

**R**can be calculated using a least squares approach. This method minimizes the quantity Ignoring the factor of 1/

*n*, expansion of Eq. (5) yields Given that all

**x**

*and*

_{i}**y**

*are fixed, minimizing Eq. (6) is therefore equivalent to maximizing where*

_{i}**C**is the cross-dispersion matrix calculated from the sum of the outer products of

**y**

*and*

_{i}**C**such that where

**W**is a diagonal matrix containing the singular values of

**C**, and

**U**and

**V**are orthogonal matrices. As was shown in [34

34. J. Challis, “A procedure for determining rigid body transformation parameters,” J. Biomech. **28**, 733–737 (1995). [CrossRef] [PubMed]

**R**is given by This procedure is applicable to all non-colinear sets of points with

*n*≥ 3.

### 2.2. Application to colloidal clusters

15. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

16. A. D. Dinsmore, E. R. Weeks, V. Prasad, A. C. Levitt, and D. A. Weitz, “Three-dimensional confocal microscopy of colloids,” Appl. Opt. **40**, 4152–4159 (2001). [CrossRef]

**x**′

*for every particle*

_{i}*i*over a distinct set of times. Hence, the first step in tracking rotational motion of a cluster is to track the translational motion of each particle within the cluster. For each cluster, we first determine the center of mass

**x**

*at a given time and subtract this quantity from the coordinates of particles belonging to the cluster, thereby removing any translational motion. We are left with new coordinates*

_{CM}**x**

*in the center of mass frame, This step is equivalent to setting the elements of*

_{i}**v**to zero in Eq. (1). With translational motion removed, we may apply Eq. (2) with a slightly different interpretation. Rather than representing a transformation between coordinate frames, we may understand

**R**as describing the rotational trajectory of a particle with inital position

**x**

*such that*

_{i}**C**and subsequently perform a singular value decomposition to calculate a rotation matrix for each pair of successive times [

*t*,

*t*+ Δ

*t*]. With the complete set of rotation matrices, {

**R**

*}, we may reconstruct the entire trajectory of a particle about the cluster center of mass by computing the product of successive rotations. Given*

^{k}*t*can be calculated as where the index

*k*enumerates the rotation between successive times.

**R**} is that it describes the collective behaviors of particles within a cluster, rather than a property of any individual particle. For example, knowledge of {

^{k}**R**} for a cluster allows for immediate calculation of the motions of any particular particle about the center of mass, or the motion of the cluster about any arbitrary axis of rotation. Diffusive anisotropic clusters with large aspect ratios rotate more freely about a long axis than about a short axis. Given {

^{k}**R**}, however, one needs only the initial orientation of these axes to compute and compare the motions around them. An additional advantage of this procedure is that it is computationally fast, and the necessary linear algebra routines are standardly included in most mathematical software suites.

^{k}## 3. Tests of the prescribed method

*N*of a pixel, where

*N*is the width of the object in pixels. Camera resolution varies between experimental set-ups but is typically in the range of 0.2

*μ*m/pixel. The minimum uncertainty in particle position is the product of these factors. For example, observing a 10 pixel wide object with a resolution of 0.2

*μ*m/pixel leads to a lower limit of ≈ 20 nm uncertainty in particle position. Other sources of noise, such as stray light entering the microscope, noise within the CCD camera itself, etc., slightly increase the uncertainty in particle position and further limit particle tracking resolution.

*D*, and different levels of noise,

_{R}*σ*. We first generate noise-free cluster trajectories. For tetrehedra, we place particles at initial coordinates where

_{x}*R*is the distance from a particle center to the cluster center of mass. In this work, we study a range of

*R*∈ [√2,

*μ*m. For pentahedra, we use initial coordinates Once initialized, we evolve each simulation for 10

^{4}time steps.

*α*,

*β*,

*γ*, from a Gaussian distribution with a standard deviation of

*γ*about the

*z*-axis, then by an angle

*β*about the

*y*-axis, and finally by angle

*α*about the

*x*-axis to produce the tetrahedron at the subsequent time. The rotation matrices used are

**x**

^{0}, the subsequent position vector is

**x**=

**R**

_{x}**R**

_{y}**R**

_{z}**x**

^{0}.

*D*. However, using the publicly available package HYDRO++ [35], we calculate the difference in

_{R}*D*between long and short axes to be ≈ 2%. Given this small variation, the use of a single

_{R}*D*to describe a pentahedron is a more than sufficient approximation for the purpose of testing our algorithm, which after all does not require that the motions agree with the laws of diffusion.

_{R}*σ*, to the particle coordinates. The levels of noise presented here correspond to uncertainties of

_{x}*σ*∈ {10, 30, 50, 70, 100} nm in the

_{x}*x*-,

*y*-, and

*z*-directions. Experimental uncertainties are typically within the range of 20-60 nm, and so the levels studied here are relevant to microscopy experiments. After the noise is added, we apply our method of measuring rotational motion in order to gauge the effect of experimental noise on results.

## 4. Analysis of rotational motion

36. M. G. Mazza, N. Giovambattista, F. W. Starr, and H. E. Stanley, “Relation between rotational and translational dynamic heterogeneities in water,” Phys. Rev. Lett. **96**, 057803 (2006). [CrossRef] [PubMed]

37. M. G. Mazza, N. Giovambattista, H. E. Stanley, and F. W. Starr, “Connection of translational and rotational dynamical heterogeneities with the breakdown of the Stokes–Einstein and Stokes–Einstein–Debye relations in water,” Phys. Rev. E **76**, 031203 (2007). [CrossRef]

**R**and can then be used to calculate the magnitude of an angular displacement. While a mathematically direct and general approach, it describes only the average cluster dynamics and can provide no insight into how motions about different axes vary. Given these limitations, we relegate a description of this method to the appendix.

**p̂**

^{0}fixed to the cluster. We determine its orientation at a later time

*t*by applying the set of rotations such that in a similar way as in Eq. (13). Although there are no constraints on what one may select

**p̂**

^{0}to be, some choices may be more enlightening than others. For example, the cluster

*n*= 5 shown in Fig. 1 has distinct long and short axes, and so one expects slower diffusion about the short axis. To quantify how rotational dynamics about these axes differ, one could choose two

**p̂**

^{0}to study separately: one choice of

**p̂**

^{0}perpendicular to the long axis; a second perpendicular to the short. Such a procedure would yield information relating to motions about the long and short axes, respectively.

36. M. G. Mazza, N. Giovambattista, F. W. Starr, and H. E. Stanley, “Relation between rotational and translational dynamic heterogeneities in water,” Phys. Rev. Lett. **96**, 057803 (2006). [CrossRef] [PubMed]

38. S. Kämmerer, W. Kob, and R. Schilling, “Dynamics of the rotational degrees of freedom in a supercooled liquid of diatomic molecules,” Phys. Rev. E **56**, 5450–5461 (1997). [CrossRef]

*t*]. The vector

*ω⃗*(

*t*′)

*dt*′ has a direction given by

**p̂**(

*t*′) ×

**p̂**(

*t*′ +

*dt*′) and magnitude |

*ω⃗*(

*t*′)

*dt*′| = cos

^{−1}[

**p̂**(

*t*′) ·

**p̂**(

*t*′ +

*dt*′)], which is the angle subtended by

**p̂**during this time interval.

*t*), consider an object with constant angular velocity

*ω⃗*=

*ω*

**ẑ**. Over a time Δ

*t*, the rotational vector displacement is given by φ⃗ (Δ

*t*) =

*ω⃗*Δ

*t*= (0, 0,

*ω*Δ

*t*). Therefore in general, φ⃗(

*t*) has components in each of the Cartesian axes, (φ

*, φ*

_{x}*, φ*

_{y}*), corresponding to cumulative rotations about those axes.*

_{z}**p̂**

^{0}through the rotation space described above. As shown in Fig. 2, even though the cluster is a solid body, trajectories of individual particles differ due to rotations of the cluster about random axes.

*t*) in Eq. (18), we may define an unbounded mean square angular displacement (MSAD), akin to a translational mean square displacement, as where the angle brackets indicate an average over all equivalent lag times Δ

*t*. In three dimensions the Stokes-Einstein-Debye relation states that the MSAD grows as where

*D*is the rotational diffusion coefficient. In this paper, we focus on measuring the MSADs of clusters, but we point out that other techniques exist to quantify rotational dynamics. For example, by observing the decay of an orientational correlation function 〈

_{R}**p̂**(

*t*+ Δ

*t*) ·

**p̂**(

*t*)〉 one can measure

*D*[39

_{R}39. B. J. Berne, P. Pechukas, and G. D. Harp, “Molecular reorientation in liquids and gases,” J. Chem. Phys. **49**, 3125–3129 (1968). [CrossRef]

40. G. Williams, “Time-correlation functions and molecular motions,” Chem. Soc. Rev. **7**, 89–131 (1978). [CrossRef]

**p̂**(

*t*) as in Eq. (17).

^{−4}rad

^{2}/ts and 10

^{−3}rad

^{2}/ts respectively and different levels of noise. The influence of noise is apparent in Fig. 3 as deviations from linearity at small Δ

*t*. Eventually, the MSAD recovers the true diffusive behavior because the cluster has made rotations large enough to distinguish from the noise.

*t*will be a straight line with a slope of unity, as indicated by the open circles in Figs. 3(a) & (b). However, in the presence of noise in particle positions, rotations cannot be accurately resolved below a certain threshold, Φ. For example, a stationary cluster will appear to make small, but fictional, rotations as a result of this noise, and measurements of the MSAD will yield where Φ

^{2}is independent of Δ

*t*due of the lack of correlations in noise.

**179**, 298–310 (1996). [CrossRef]

*R*

^{2}, where

*R*is the average distance from a particle to the cluster center of mass. Our matrix method reduces this uncertainty further by a factor of

*n*. Thus, when only static noise is present, one expects for a diffusing cluster where

*D*= 0) tetrahedral clusters and calculate the MSADs. As in Eq. (21), the MSADs are constant over time. We take Φ

_{R}^{2}to be the value of the MSAD as Δ

*t*→ 0. Shown in Fig. 4(a), Eq. (23) accurately describes the static angular uncertainty for a wide range of tetra- and pentahedral cluster sizes,

*R*, and noise levels. Fig. 4(b) shows all of the previously measured MSADs in Fig. 3 plotted with the noise subtracted. This precisely collapses the MSADs to the true values in each case.

*σ*, the diffusion constant

_{x}*D*and calculating an approximate Φ

_{R}^{2}. Diffusive motion will begin exceeding noise when Φ

^{2}≈ 4

*D*Δ

_{R}*t*. Solving for Δ

*t*then yields a reasonable sampling lag time. It is also important to avoid under-sampling, that is, too long a lag time. Doing so will make diffusive motion between images appear erroneously slow. We find that a good rule of thumb for the upper limit on sampling time should be the time when the cluster has diffused ≈1 radian

^{2}. Thus, an estimate for the upper limit is ≈ 1/4

*D*. However, as in all particle tracking experiments, the time between images must be small enough that individual particles can be confidently identified. This typically means that particles must be imaged before moving a distance of one interparticle spacing [15

_{R}**179**, 298–310 (1996). [CrossRef]

## 5. Experimental application

29. M. T. Elsesser, A. D. Hollingsworth, K. V. Edmond, and D. J. Pine, “Large core-shell poly(methyl methacrylate) colloidal clusters: synthesis, characterization, and tracking,” Langmuir **27**, 917–927 (2011). [CrossRef]

29. M. T. Elsesser, A. D. Hollingsworth, K. V. Edmond, and D. J. Pine, “Large core-shell poly(methyl methacrylate) colloidal clusters: synthesis, characterization, and tracking,” Langmuir **27**, 917–927 (2011). [CrossRef]

*μ*m as measured by static light scattering (SLS). The particles within a cluster are irreversibly bound together, but are sterically stabilized to prevent the possibility of aggregation to other clusters. Dilute suspensions of clusters are prepared in a mixture of cyclohexyl bromide (CXB) and

*cis*-decalin (DCL) at a ratio of 85/15 (w/w) that closely matches both the density and index of refraction of the particles. Clusters are imaged in 3D over time with a Leica TCS SP5 confocal microscope. We track locations of the individual particles within a tetrahedron using standard particle tracking routines [15

**179**, 298–310 (1996). [CrossRef]

**40**, 4152–4159 (2001). [CrossRef]

*x*- and

*y*-directions, and ≈ 40 nm in the

*z*-direction. Given these tracking resolutions and assuming a maximally packed tetrahedron, from Eq. (23) we estimate the angular resolution in this experiment as Φ ≈ 0.028 radians (1.6°).

*D*and

_{T}*D*, respectively. In three dimensions, the translational MSD is described by the Stokes-Einstein-Sutherland equation while the MSAD is described by Eq. (20).

_{R}41. M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D Brownian diffusion of submicron-sized particle clusters,” ACS Nano **3**, 3326–3334 (2009). [CrossRef] [PubMed]

*d*

_{tetra}is the effective hydrodynamic diameter of the cluster, and

*d*

_{sphere}is the diameter of the particles within the cluster. Theoretical translational and rotational diffusion coefficients,

*D*and

_{T}*D*respectively, can be calculated using where

_{R}*k*is Boltzmann’s constant,

_{B}*T*is the temperature, and

*η*is the viscosity of the suspending solvent. The viscosity of the CXB/DCL mixture was measured at

*η*= 2.18 mPas and experiments were performed at

*T*= 295 K.

**p̂**

^{0}). Solid lines are fits to the data over the indicated range 3 s ≤ Δt ≤ 22 s. As can be seen, both the MSD and MSAD are approximately linear only at small lag times. This is because the data set used to make these measurements consists of only a single cluster for less than 300 timesteps (≈ 220 s). Diffusion coefficients are measured by performing linear fits to the MSD and MSAD shown in Fig. 5 over the indicated range and extracting the slopes of the lines.

*d*

_{tetra}are calculated from Eqs. (26) & (27) using known experimental conditions, and values for

*d*

_{sphere}then follow trivially using Eq. (25). Ideally, the sizes calculated from translational motions would be identical to those calculated from rotational motions. As shown, these values agree to within 4%. Such good agreement between these two measurements demonstrates our ability to track translational and rotational motions of clusters simultaneously.

*μ*m measured by SLS [29

**27**, 917–927 (2011). [CrossRef]

## 6. Conclusion

*R*. When combined, the resolution of measuring angular displacements scales with 1/

*nR*

^{2}.

## Appendix

**R**, one can calculate at each time the axis of rotation

**û**and angular displacement Δφ. In this notation, a rotational displacement can be described by a vector Δφ⃗ = Δφ

**û**, where

**û**has components in each of the Cartesian axes.

**û**, therefore, From Eq. (28), we see that the axis of rotation is an eigenvector of the matrix

**R**with an eigenvalue of 1. For a set of rotational displacements {

**R**

*}, one may determine the axes of rotation by calculating the eigenvectors and eigenvalues of the rotation matrices, searching for the eigenvalues equal to 1, and taking the corresponding eigenvectors.*

^{k}**ŵ**perpendicular to

**û**. For simplicity, we choose

**ŵ**to be perpendicular to the

*x*-axis (denoted by

**î**), and apply the rotation matrix, The magnitude of the displacement is the angle between

**ŵ**and

**ŵ**′, and can be computed using the cross product relation Calculating displacements relative to the axis of rotation always results in displacements greater than or equal to those measured relative to an arbitrary

**p̂**

^{0}. For example, measuring a diffusion coefficient of a spherically symmetric body with this method will yield a value that is a factor of 3/2 of the actual diffusion coefficient in Eq. (20).

**û = ẑ**, a perpendicular vector

**ŵ**=

**x̂**, and an arbitrary vector

**p̂**

^{0}located at Cartesian coordinates (sin

*θ*, 0, cos

*θ*), where

*θ*is the spherical polar angle. If

**ŵ**is rotated by an amount

*δ*φ

*, the angle between*

_{u}**ŵ**and

**ŵ**′ is identically

*δ*φ

*. However, the angle*

_{u}*δ*φ between

**p̂**and

**p̂**′ can be shown to be

*δ*φ

*, we can approximate*

_{u}*δ*φ as

## Acknowledgments

## References and links

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35. | HYDRO++, http://leonardo.inf.um.es/macromol/programs/hydro++/hydro++.htm. |

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39. | B. J. Berne, P. Pechukas, and G. D. Harp, “Molecular reorientation in liquids and gases,” J. Chem. Phys. |

40. | G. Williams, “Time-correlation functions and molecular motions,” Chem. Soc. Rev. |

41. | M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D Brownian diffusion of submicron-sized particle clusters,” ACS Nano |

42. | F. Perrin, “Étude mathématique du mouvement Brownien de rotation,” Ann. Sci. Ec. Normale Super. |

**OCIS Codes**

(000.2170) General : Equipment and techniques

(000.4430) General : Numerical approximation and analysis

(100.2000) Image processing : Digital image processing

(180.1790) Microscopy : Confocal microscopy

(180.6900) Microscopy : Three-dimensional microscopy

(100.4999) Image processing : Pattern recognition, target tracking

**ToC Category:**

Microscopy

**History**

Original Manuscript: July 13, 2011

Revised Manuscript: August 4, 2011

Manuscript Accepted: August 8, 2011

Published: August 17, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Gary L. Hunter, Kazem V. Edmond, Mark T. Elsesser, and Eric R. Weeks, "Tracking rotational diffusion of colloidal clusters," Opt. Express **19**, 17189-17202 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17189

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